The Involution Kernel and the Dual Potential for Functions in the Walters’ Family

First, we set a suitable notation. Points in {0,1}Z-{0}={0,1}N×{0,1}N=Ω-×Ω+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{0,1\}^{{\mathbb {Z}}-\{0\}} =\{0,1\}^{\mathbb {N}}\times \{0,1\}^{\mathbb {N}}=\Omega ^{-} \times \Omega ^{+}$$\end{document}, are denoted by (y|x)=(...,y2,y1|x1,x2,...)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$( y|x) =(...,y_2,y_1|x_1,x_2,...)$$\end{document}, where (x1,x2,...)∈{0,1}N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_1,x_2,...) \in \{0,1\}^{\mathbb {N}}$$\end{document}, and (y1,y2,...)∈{0,1}N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(y_1,y_2,...) \in \{0,1\}^{\mathbb {N}}$$\end{document}. The bijective map σ^(...,y2,y1|x1,x2,...)=(...,y2,y1,x1|x2,...)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\sigma }}(...,y_2,y_1|x_1,x_2,...)= (...,y_2,y_1,x_1|x_2,...)$$\end{document} is called the bilateral shift and acts on {0,1}Z-{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{0,1\}^{{\mathbb {Z}}-\{0\}}$$\end{document}. Given A:{0,1}N=Ω+→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A: \{0,1\}^{\mathbb {N}}=\Omega ^+\rightarrow {\mathbb {R}}$$\end{document} we express A in the variable x, like A(x). In a similar way, given B:{0,1}N=Ω-→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B: \{0,1\}^{\mathbb {N}}=\Omega ^{-}\rightarrow {\mathbb {R}}$$\end{document} we express B in the variable y, like B(y). Finally, given W:Ω-×Ω+→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W: \Omega ^{-} \times \Omega ^{+}\rightarrow {\mathbb {R}}$$\end{document}, we express W in the variable (y|x), like W(y|x). By abuse of notation, we write A(y|x)=A(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(y|x)=A(x)$$\end{document} and B(y|x)=B(y).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(y|x)=B(y).$$\end{document} The probability μA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _A$$\end{document} denotes the equilibrium probability for A:{0,1}N→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A: \{0,1\}^{\mathbb {N}}\rightarrow {\mathbb {R}}$$\end{document}. Given a continuous potential A:Ω+→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A: \Omega ^+\rightarrow {\mathbb {R}}$$\end{document}, we say that the continuous potential A∗:Ω-→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^*: \Omega ^{-}\rightarrow {\mathbb {R}}$$\end{document} is the dual potential of A, if there exists a continuous W:Ω-×Ω+→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W: \Omega ^{-} \times \Omega ^{+}\rightarrow {\mathbb {R}}$$\end{document}, such that, for all (y|x)∈{0,1}Z-{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(y|x) \in \{0,1\}^{{\mathbb {Z}}-\{0\}}$$\end{document}A∗(y)=A∘σ^-1+W∘σ^-1-W(y|x).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A^* (y) = \left[ A \circ {\hat{\sigma }}^{-1} + W \circ {\hat{\sigma }}^{-1} - W \right] (y|x). \end{aligned}$$\end{document}We say that W is an involution kernel for A. It is known that the function W allows to define a spectral projection in the linear space of the main eigenfunction of the Ruelle operator for A. Given A, we describe explicit expressions for W and the dual potential A∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^*$$\end{document}, for A in a family of functions introduced by P. Walters. Denote by θ:Ω-×Ω+→Ω-×Ω+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta : \Omega ^{-} \times \Omega ^{+} \rightarrow \Omega ^{-} \times \Omega ^{+}$$\end{document} the function θ(...,y2,y1|x1,x2,...)=(...,x2,x1|y1,y2,...).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta (...,y_2,y_1|x_1,x_2,...)= (...,x_2,x_1|y_1,y_2,...).$$\end{document} We say that A is symmetric if A∗(θ(x|y))=A(y|x)=A(x).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^* (\theta (x|y))= A(y|x)= A(x).$$\end{document} We present conditions for A to be symmetric and to be of twist type. It is known that if A is symmetric then μA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _A$$\end{document} has zero entropy production.